Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have the formula (part of the formulation for a homography):

$R*(n^T*x_p*I-t*n^T)$

and I don't understand how it can work given:

  • $R$ is size 3x3
  • $n^T$ is size 3x1
  • $x_p$ is size 3x1
  • $I$ is size 3x3
  • $t$ is size 3x3

The result should be a 3x3 matrix, but I cannot multiply a $x_p$ (3x1) with $I$ (3x3) for starters? Is there a different precedence? If a result is a 1x1 matrix should I consider it a scalar?

Thanks

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Since $n^Tx_p$ is a (1x1) scalar, $n^T \cdot x_p \cdot I = (n^Tx_p)I$ is a scalar multiple of the identity matrix.

Now we have that $n^Tx_pI$ is a (3x3) matrix while $tn^T$ is a (3x1) vector. Are you sure $t$ isn't a scalar?

EDIT: If $n^T$ is (3x1) and $x_p$ is (3x1), then $n^Tx_p$ doesn't make sense. Did you mean for $n$ to be (3x1) so that $n^T$ is (1x3)?

EDIT 2: Here is a consistent set of dimensions (which is hopefully correct)

  • $R$ is a (3x3) rotation matrix

  • $n$ is a (3x1) vector, so $n^T$ is (1x3)

  • $x_p$ is a (3x1) vector

  • $I$ is the (3x3) identity matrix

  • $t$ is a (3x1) vector

The dimensions should work out now.

share|improve this answer
    
Thanks, so a 1x1 is a scalar. $t*N^T$ would be a scalar too then? And then it would be an element-by-element subtraction from the scalar multiple of the identity matrix? –  aledalgrande Aug 5 at 2:03
    
Actually the right part ($t*N^T$) gives me a 3x3, but it is a mistery to me why... –  aledalgrande Aug 5 at 2:07
    
Just saw your edit: the $T$ superscript means transpose, so your first version makes sense. $t$ is a 3D translation, so safe to assume it's a 3x1. –  aledalgrande Aug 5 at 2:08
    
Yes, that is the correct set of dimensions. The mystery is still $t*n^T$ which means (3x1) * (1x3), how does that work? In the matrix library I use, the result is 3x3, but can't understand why? –  aledalgrande Aug 5 at 2:47
    
How is that a mystery? The inner dimensions match, and the result is the outer dimensions i.e. (3x1)*(1x3) yields a (3x3) matrix. –  JimmyK4542 Aug 5 at 2:52

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.